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Matrix completion of a partially revealed 5 by 5 matrix with rank-1. Left: observed incomplete matrix; Right: matrix completion result. Matrix completion is the task of filling in the missing entries of a partially observed matrix, which is equivalent to performing data imputation in statistics. A wide range of datasets are naturally organized ...
For a commutative ring and an element , a matrix factorization of is a pair of n-by-n matrices , such that =. This can be encoded more generally as a /-graded ...
Decomposition: = where C is an m-by-r full column rank matrix and F is an r-by-n full row rank matrix; Comment: The rank factorization can be used to compute the Moore–Penrose pseudoinverse of A, [2] which one can apply to obtain all solutions of the linear system =.
In mathematics, a matrix factorization of a polynomial is a technique for factoring irreducible polynomials with matrices. David Eisenbud proved that every multivariate real-valued polynomial p without linear terms can be written as AB = pI , where A and B are square matrices and I is the identity matrix . [ 1 ]
Let A be a square n × n matrix with n linearly independent eigenvectors q i (where i = 1, ..., n).Then A can be factored as = where Q is the square n × n matrix whose i th column is the eigenvector q i of A, and Λ is the diagonal matrix whose diagonal elements are the corresponding eigenvalues, Λ ii = λ i.
A common choice is to use the sparsity pattern of A 2 instead of A; this matrix is appreciably more dense than A, but still sparse over all. This preconditioner is called ILU(1). One can then generalize this procedure; the ILU(k) preconditioner of a matrix A is the incomplete LU factorization with the sparsity pattern of the matrix A k+1.
The QR factorization of a matrix is a matrix and a matrix so that A = QR, where Q is orthogonal and R is upper triangular. [1]: 50 [4]: 223 The two main algorithms for computing QR factorizations are the Gram–Schmidt process and the Householder transformation.
In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced / ʃ ə ˈ l ɛ s k i / shə-LES-kee) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for efficient numerical solutions, e.g., Monte Carlo simulations.